Optimal slice thickness for object detection with longitudinal partial volume effects in computed tomography

Abstract Longitudinal partial volume effects (z‐axial PVE), which occur when an object partly occupies a slice, degrade image resolution and contrast in computed tomography (CT). Z‐axial PVE is unavoidable for subslice objects and reduces their contrast according to their fraction contained within the slice. This effect can be countered using a smaller slice thickness, but at the cost of an increased image noise or radiation dose. The aim of this study is to offer a tool for optimizing the reconstruction parameters (slice thickness and slice spacing) in CT protocols in the case of partial volume effects. This optimization is based on the tradeoff between axial resolution and noise. For that purpose, we developed a simplified analytical model investigating the average statistical effect of z‐axial PVE on contrast and contrast‐to‐noise ratio (CNR). A Catphan 500 phantom was scanned with various pitches and CTDI and reconstructed with different slice thicknesses to assess the visibility of subslice targets that simulate low contrast anatomical features present in CT exams. The detectability score of human observers was used to rank the perceptual image quality against the CNR. Contrast and CNR reduction due to z‐axial PVE measured on experimental data were first compared to numerical calculations and then to the analytical model. Compared to numerical calculations, the simplified algebraic model slightly overestimated the contrast but the differences remained below 5%. It could determine the optimal reconstruction parameters that maximize the objects visibility for a given dose in the case of z‐axial PVE. An optimal slice thickness equal to three‐fourth of the object width was correctly proposed by the model for nonoverlapping slices. The tradeoff between detectability and dose is maximized for a slice spacing of half the slice thickness associated with a slice width equal to the characteristic object width.

according to their fraction contained within the slice. This effect can be countered using a smaller slice thickness, but at the cost of an increased image noise or radiation dose. The aim of this study is to offer a tool for optimizing the reconstruction parameters (slice thickness and slice spacing) in CT protocols in the case of partial volume effects. This optimization is based on the tradeoff between axial resolution and noise. For that purpose, we developed a simplified analytical model investigating the average statistical effect of z-axial PVE on contrast and contrast-to-noise ratio (CNR). A Catphan 500 phantom was scanned with various pitches and CTDI and reconstructed with different slice thicknesses to assess the visibility of subslice targets that simulate low contrast anatomical features present in CT exams. The detectability score of human observers was used to rank the perceptual image quality against the CNR. Contrast and CNR reduction due to z-axial PVE measured on experimental data were first compared to numerical calculations and then to the analytical model. Compared to numerical calculations, the simplified algebraic model slightly overestimated the contrast but the differences remained below 5%. It could determine the optimal reconstruction parameters that maximize the objects visibility for a given dose in the case of z-axial PVE. An optimal slice thickness equal to three-fourth of the object width was correctly proposed by the model for nonoverlapping slices. The tradeoff between detectability and dose is maximized for a slice spacing of half the slice thickness associated with a slice width equal to the characteristic object width.

| INTRODUCTION
The detection of thin objects with low contrast in computed tomography (CT) is crucial to differentiate anatomical features with close densities and distinguish tumors. Consequently, studies regularly investigate the best settings to improve object detectability for different tasks. 1-3 Contrast and detection of small objects can be dramatically lowered when spatial resolution along the longitudinal z-axis is not adapted to object thickness. Z-axial (longitudinal) partial volume effect (PVE) is unavoidable for subslice objects (objects thinner than the slice width). It can also randomly affect those smaller than twice the slice thickness according to their fraction contained within the slice. Object undersampling in the longitudinal direction introduces quantitative biases in Hounsfield units (HU), and adversely lowers the contrast. [4][5][6] Contrast loss introduced by z-axial PVE is not routinely considered and is often overlooked in clinical practice since it can be reduced using thin-slice scanning. Reconstructing thin slices is, however, associated with small pitches, high noise, or increased dose. Nevertheless, tradeoffs between axial resolution and noise can be envisaged to optimize CT protocols for an envisaged clinical application. 7 Previous studies have tested various protocols to find the slice thickness, 8-10 slice spacing, 11,12 or interpolation algorithm 13 that would give the best compromise between z-axial resolution and noise. Although the physical relationships between axial resolution and noise in CT are long known, the stochastic influence of z-axial PVE on the contrast of small objects has never been taken into account. Clinical medical physicists have, however, to deal with partial volume effects for (1) optimizing the objects detectability, and (2) minimizing the dose, which is especially important in CT. 1,14 The goal of this study is to offer a tool for optimizing the reconstruction parameters (slice thickness and slice spacing) in CT protocols for a given detection task involving z-axial PVE. The z-axial PVE is a complex phenomenon that implies statistical variations due to the object positioning within the slice, and a simple signal averaging in the slice volume is not adequate for predicting contrast attenuation and detection jeopardizing. The exact determination of an expected contrast-to-noise ratio (CNR) with z-axial PVE requires numerical computation based on the object and slice z-profiles, and the optimization problem is therefore difficult to resolve in clinical practice. This study develops an approximate analytical model that allows an easy determination of the reconstruction parameters corresponding to the highest CNR or lowest dose for cross-sectional imaging of thin objects. It uses a quantitative relationship between noise, axial resolution, and dose to establish the optimal reconstruction parameters that maximize detection tasks with z-axial PVE. It averages out the stochastic object positioning within the slices, as a function of object size and slice thickness. First, it is tested using a standard image quality phantom made of well-calibrated objects, shapes, and length. Then, it is compared to the exact solution of the problem determined by numerical computation. The contrast, CNR, and detectability of thin objects of different lengths are assessed with z-axial PVE for different imaging conditions (pitch, noise, and slice thickness). A reader study confirms that the detectability of subslice targets is CNR dependent, as expected from previous results obtained for larger objects, 2,15 and validates the relevance of the algebraic model for optimal scanning parameters choice and dose minimization.
When a voxel contains different densities, the resulting signal within the voxel is the average of the signals from the different materials.
The signal (HU) for an object located partially in a slice is weighted according to its fraction within the slice. Accordingly, the contrast C(z) of such objects may vary depending on their z-position in a slice.
It is given by the convolution product between the longitudinal contrast profile C 0 (z) of the object and the slice sensitivity profile (SSP(z)), the point spread function (PSF) of the imaging system along the z-axis For the particular case of a cylindrical object of length L and CT number l located in a background of CT number l 0 , ; with the rectangular function rect x ð Þ ¼ 1; x 2 ½À0:5; 0:5 0; otherwise : The statistical expected object contrast C is given by the value of C(z) averaged over all the possible z-positions within the effective The effective slice thickness (T e ) depends on the slice sensitivity profiles (SSP) extent. The full width at half maximum (FWHM) is used for T e in usual regulations. 16,17 The SSP is, however, not fully characterized by its FWHM. The longitudinal signal spread will depend on its entire shape: long tails will degrade the z-resolution more than a close to rectangular SSP, even if both have the same FWHM. 18 For a more precise characterization of the longitudinal signal spread, the mean width of the SSP was used as a measure of the effective slice thickness in this study.

2.A.2 | Simplified algebraic model
A simplified algebraic expression for the expected contrast can be used to provide an understanding relationship between the quantitative parameters involved in the contrast loss. This is done by approximating the SSP with a rectangular function of height equal to one, and width equal to the effective slice thickness (T e ). The object contrast profile C 0 (z) can also be approximated by a rectangular function of height C 0 and width L. In this study, L represents the characteristic object length-for example, p 4 d for a sphere of diameter d. Considering the slice which contains the largest proportion of the object (the slice with the highest object contrast), the object length in this slice can be between L/2 and min(T e , L). Considering k = L/T e the ratio between the characteristic object length and the effective slice thickness, z-axial PVE may occur only if k < 2. Two cases have to be further distinguished: 1) the object is thinner than T e (0 < k ≤ 1), and z-axial PVE is unavoidable; 2) the object width is between one and two-times T e (1 ≤ k < 2), and z-axial PVE will occur stochastically, depending on the object fraction within the slice.
Still considering the slice with the largest object length (slice with the highest contrast), the expected object contrast C will be weighted by the expected object fraction within the effective slice width (L=T e ). For a homogeneous object of CT number l in a background of CT number l 0 , this gives Equation (5) shows that z-axial PVE reduces the expected object contrast by the factor k(1 À k/4) ≤ 1 (for k ≤ 2). The highest and lowest contrasts possible, always considering the slice with the highest contrast, occur when the entire object and half object are contained in the slice. They are given in eqs. (6a) and (6b), respectively.
For a slice spacing aT e (a ≤ 1), the partial overlap between adjacent slices leads to a decrease in the length over which the expected object length and contrast must be integrated (red area in Fig. 1).
The integration length is then reduced to (2a À 1)T e instead of T e .
The expected object length and contrast for 0.5 < a ≤ 1 are given in eqs. (7) and (8), respectively.
As expected, the particular case a = 1 (non-overlapping slices) in eqs. (7) and (8) leads to eqs. (4) and (5), respectively. For a between 0.5 and 1, the smaller the slice spacing the closer the expected contrast curve approaches the highest contrast possible (Fig. 2). Additionally, the range in which z-axial PVE can occur is determined by k < 2a, and decreases with a. A value of a equal to 0.5 will give the highest expected contrast for a given slice thickness. Reducing a to less than half is useless.

2.C | Effective slice thicknesses (T e )
The 0.28-mm tungsten carbide bead of the CTP528 module of the Catphan phantom was used as a subpixel point source to determine the slice sensitivity profiles (SSP) of the CT slices along the z-direction. 19 The bead was scanned using the lowest noise index (NI = 5) and two .

2.E | Subjective image quality evaluation
Phantom scoring was used to rank the perceptual image quality against the CNR, whose value is reduced by z-axial PVE.  A, a, b, and c (Fig. 3) was noted on a three level scale, 0, 0.5, and 1, corresponding to a nonvisible, partially visible, and totally visible object. The average image quality score for the 10 acquisitions of a given protocol was established for each cylinder and observer. The agreement between the four observers was assessed with the Fleiss' kappa test using the R statistical software (R Development Core Team).

3.A | Subjective image quality evaluation
The concordance level of subjective image scoring for the four human observers (j) was 0.504, and indicated an adequate interreader agreement according to the scale of Landis and Koch. 20 The average image quality scores of the four observers were therefore considered for the 18 protocols and the four cylinders. The average scores were linked to their corresponding object CNRs with a fitted sigmoid curve (Fig. 4).

3.C | Contrast reduction factor
All the FWHMs and mean widths of the measured SSP (Fig. 6) given in Table 2 (2) with the simplified algebraic model (eq. 5; Fig. 8). All the measured contrasts are at their nominal value for k ≥ 2, where no z-axial PVE occurs. But as expected, they fall off as a function of k below 2, due to z-axial PVE.
The variability in the averaged contrasts measured for a given L/T e ratio shows statistical variability in z-axial PVE between the data acquired with the three different noise indexes. Averaged normalized contrasts measured on the slices are consistent with the calculated expected contrast reduction factors given in Table 2. An exception happened for the smallest L/T e ratio (k = 0.6) for which measured contrasts were up to 50% lower than the theoretical values obtained from numerical and analytical calculations. Compared to the numerical calculation, the simplified algebraic model slightly overestimates the contrast but the differences always remain below 5%.

3.D | Contrast-to-noise ratio (CNR)
The expected contrast from the simplified model (eq. 8) and the noise level from eq. (10) were used to determine the theoretical expected CNR (CNR ) of the rods as a function of the ratio k = L/T e, The measured CNR agree with the values predicted by eq. (11), except for the smallest L/T e ratio, as measured contrasts were up to 50% lower than the theoretical values (Fig. 9). Equation (11) was used to analyze the slice thickness required to restore the CNR to an optimal value for a given CTDI. The expected CNR is the highest for k equal to that reduces to eq. 12b for slices reconstructed by FBP (b = 0.5) The measured CNR shown in Fig. 9 reach a peak for k = 4/3, as predicted by eq. (12b) for nonoverlapping slices (a = 1). The choice T A B L E 2 FWHM and mean width of the SSP, and expected contrast reduction factor obtained from eq. (2) for the three rods.     < 1), the optimal k decreases with the slice spacing and reaches 1 for a = 0.5. This implies that the optimal slice thickness increases with the slices overlap from 0.75L up to L for 50% overlap. Figure 10 shows the iso-CNR curve based on eq. (11) as a function of the ratio between the chosen and optimal slice thickness. It indicates how a suboptimal choice in slice thickness should be compensated by an increase in CTDI for maintaining a given CNR in the case of z-axial PVE. A choice of a slice two times thicker than the optimal thickness would, for example, necessitate a dose increase of 28% to hold a constant CNR for nonoverlapping slices. Figure 11 gives the optimal parameter k as a function of slice spacing and noise behavior (parameter b). For nonoverlapping slices reconstructed with iterative algorithms, the optimal slice thickness is thin- Weber's law. This law predicts the detection probability of a given object as a function of the stimulus magnitude on the image follows a psychometric curve described by the integral of the normal probability curve. 24 In our article we used cylindrical rods to represent for instance tumors of different sizes. If the rods were placed at different orientations it would have change two aspects. Indeed, a change in object orientation relative to the longitudinal CT z-axis reverses to a change in the characteristic longitudinal object length L. It will also affect the sharpness of the object boundaries. These two effects are considered in our model through the calculation of the characteristic object length (L), defined as the average width of the object z-profile.

Slice thickness (mm) Pitch
A modification in the orientation of the objects would shorten L, and thereby the value of the variable k = L/T e considered in our study.
The optimal slice thickness given by the model would thus change with the object orientation in the same way it changes with the object length or size.
Given the simplicity of the objects' shape and phantom's geometry used in our study, the algebraic model was tested under experi- This may be the scope of a future work that simulates various tumor shapes in tissue-like surroundings.
Nevertheless, the analytical model may be easily applied for a complicated clinical task such as liver lesion detection. For example, Soo et al 8 found that a slice thickness of 5 mm allowed a better detection, by radiologists, of lesions <5 mm in diameter than 2.5, 7.5, and 10 mm slices. Z-axial PVE dramatically decreased the detection rate from 95% for 5 mm slices to 65% and 42% for the thicker 7.5-and 10-mm slices, respectively, while an increase in image noise on the thinner 2.5-mm slices decreased detection to 90%. An optimal slice thickness is therefore expected between 2.5 and 5 mm for lesions~5 mm in diameter. Our model applied to this task gives an optimal slice thickness of T ffi 0.6d = 3 mm for spherical lesions of

| CONCLUSION
An analytical model for the average statistical effect of z-axial PVE on contrast-to-noise ratio (CNR) and detection was developed, and allowed to determine the optimal slice thickness and spacing for a given detection task. Z-axial PVE was confirmed to significantly degrade the detection of thin objects in CT imaging, causing a loss in CNR. The analytical model provides an understanding relationship between axial resolution, noise, and dose that takes into account the expected contrast falloff due to z-axial PVE. The averaged contrasts and CNR measured on the slices for different object sizes and positioning were in a good agreement with the model. The model allows knowledgeable selection of the reconstruction parameters (slice thickness and spacing) that, for a given z-axial PVE, are required to restore the CNR to values that maximize the tradeoff between resolution and noise, or minimize the dose for a given object detection task. This model will contribute to help optimize CT protocols.

ACKNOWLEDG MENTS
The authors gratefully acknowledge the four readers who spent time reading and scoring the objects visibility on CT slices.

CONFLI CT OF INTEREST
The authors declare no conflict of interest.

R E F E R E N C E S
1. Funama Y, Awai K, Nakayama Y, et al. Radiation dose reduction without degradation of low-contrast detectability at abdominal